About one approximate discrete-continual theory of a plane stress state | Mekhanika | kompozitsionnykh | materialov i konstruktsii
> Volume 25 > №3 / 2019 / Pages: 336-353

About one approximate discrete-continual theory of a plane stress state


An approximate linear discrete-continuum analytical theory described by differential-difference equations is constructed for a rectangular elastic isotropic plate of constant thickness using the finite element method. The plate was represented by a regular discrete-one-dimensional elastic system composed of the same rectangular elements in the plan. Their length coincided with one of the dimensions of the plate, and the transverse size was determined by the ratio of the other size of the plate to the specified number of elements. The plane stress state was taken as the initial model of deformation of the plate and finite elements. The displacements of all the finite elements were approximated in the transverse direction linearly so that the geometric conditions of conjugation of the adjacent elements were satisfied. This made it possible, at the initial stage, to reduce the two-dimensional continuum theory of a plane stress state to a discrete-continuum theory described by functions of two arguments. One of them is the continual variable (the longitudinal Cartesian coordinate), and the other is an integer parameter, by means of which the elements of the elastic system are numbered. A rigorous discrete-continual analysis of the finite-element elastic system based on the gluing method and the variational principles of Lagrange and Castigliano allowed us to reveal generalized displacements, deformations, internal and external forces of the theory under study and establish its defining geometric, physical and static relationships, including the equations of compatibility of deformations. Within the framework of the theory, alternative formulations of differential-difference boundary-value problems in generalized displacements and internal forces are given. When the problem is formulated in the internal forces, force functions (discrete-continual analogs of stress functions) are introduced, which have made it possible to reduce the number of differential-difference resolving equations. The application of the theory is illustrated on a flat rod for which the simplest statically indeterminate model of deformation is constructed, with Bernoulli and Tymoshenko models as special cases.